arXiv subjects

Xiang Zhong

Publications and source records attributed to Xiang Zhong.

12 recordsLinked to original sources

arXiv2026

A time explicit multiscale framework for heterogeneous elastic wave propagation based on multipoint stress discretization

Simulating elastic wave propagation in heterogeneous media presents significant mathematical and computational challenges, since resolving fine-scale material variations requires extremely fine spatial discretizations, while mixed stress--displacement formulations typically lead to large saddle-point systems that are not well suited for efficient time integration. Moreover, conventional multiscale reductions generally produce nontrivial coarse mass matrices, thereby requiring additional linear solves at every time step and diminishing the advantages of explicit schemes. To address these difficulties, we develop a time explicit multiscale method based on a multipoint stress control volume discretization. The central innovation is a unified spatial--temporal reduction strategy that simultaneously removes two dominant algebraic bottlenecks: the complex global fine-scale mixed solve and the repeated coarse-scale mass solve arising in conventional multiscale time stepping. More specifically, the stress and auxiliary rotation variables are eliminated locally, leading to a symmetric positive definite reduced stiffness operator, upon which a multiscale space is constructed using a density-weighted local spectral problem. The associated projected mass bilinear form preserves the physical mass inner product and gives an identity mass matrix under a density-orthonormal auxiliary basis, leading naturally to an explicit central-difference scheme. We establish discrete energy stability under a coarse-scale CFL condition and derive convergence estimates for both the density-weighted displacement and the locally recovered stress. Numerical experiments in heterogeneous media validate the theoretical results and demonstrate the effectiveness of the proposed method for elastic wave propagation.

math.NA↗

arXiv2026

Multiscale Modeling for Time-harmonic Maxwell equations with impedance boundary conditions in highly heterogeneous media

Modeling time-harmonic Maxwell problems in heterogeneous media presents significant mathematical and computational challenges. Due to the inherent non-elliptic structure and non-coercive nature of Maxwell equations, conventional methods face severe numerical instabilities, particularly in high-contrast media and at high wave numbers. These challenges often lead to ill-conditioned discrete systems and prohibitively high computational costs, limiting their practical applicability. To overcome these challenges, we introduce an efficient multiscale framework for time-harmonic Maxwell equations with impedance boundary conditions in high-contrast media. A major novelty of this study lies in circumventing the need for an explicit divergence-free constraint on multiscale basis functions. To achieve this, an auxiliary space is constructed via local spectral problems incorporating a mass term and a Silver-Müller-type boundary penalty. This novel design guarantees the coercivity of the corresponding bilinear form and automatically excludes the kernel of the curl operator from the leading eigenspaces. Building upon the auxiliary space, we then construct the multiscale space by using a distinct bilinear form. By exploiting a resolution condition and establishing key norm relationships, we rigorously prove the coercivity of this modified bilinear form a crucial property that underpins the whole analysis. Theoretical analysis shows that, with appropriate oversampling, the method achieves $O (H)$ convergence independent of the local contrast and the approximation error increases with the wave number $k$. Extensive numerical experiments are reported to validate the effectiveness of the proposed approach.

math.NA↗

arXiv2026

A Unified Framework for Structure-Aware Clustering and Heterogeneous Causal Graph Learning

In complex multivariate systems, interactions among variables are defined by dependency structures, often encoded as directed acyclic graphs ($\text{DAGs}$). However, dependency structures can vary across subjects, and ignoring this structural heterogeneity introduces bias and obscures subpopulation-specific dependencies. To address this, we propose Directed Acyclic Graph-based Dependency Clustering via Alternating Direction Method of Multipliers (DAG-DC-ADMM), a unified framework built upon Structural Equation Modeling (SEM) that jointly learns cluster assignments and cluster-specific dependency structures. We encode acyclicity via a smooth constraint and integrate a groupwise truncated Lasso fusion penalty (gTLP) to cluster subjects based on their structural similarity. This yields a nonconvex optimization problem that incorporates sparsity, acyclicity, and structural consensus constraints. We address the nonconvexity by using the augmented Lagrangian method and solve it with an adapted version of the Alternating Direction Method of Multipliers (ADMM) for difference-of-convex programs. For certain graph structures, such as upper triangular adjacency matrices, our algorithm is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) point. Experiments demonstrate that our method recovers cluster-specific causal dependency structures with a high true positive rate and a low false discovery rate. This capability enables the robust discovery of heterogeneous dependencies across subjects where the subpopulation label is unknown.

stat.ML↗

arXiv2026

Efficient Multiscale Methods for Highly Heterogeneous Spatial Network Models

Modeling complex spatial networks with multiscale heterogeneity poses significant mathematical and computational challenges. Lacking explicit PDE discretizations and facing excessive degrees of freedom, conventional methods often become computationally prohibitive. To address these challenges, we propose an efficient multiscale modeling for highly heterogeneous spatial networks. We construct multiscale basis functions tailored to spatial network models with heterogeneous edge weights and node degrees. A key novelty is that the proposed method doesn't introduce geometric parameters (such as Dirichlet nodes, distances, or mesh sizes), thereby preserving its purely algebraic nature and ensuring broad applicability. By incorporating a subgraph-wise estimate, we define a Poincaré constant $C_{\mathrm{po}}$ that renders the method independent of the underlying graph geometry. Then through an appropriate choice of the number of graph oversampling layers, we establish an $O(C_{\mathrm{po}})$ convergence independent of the local heterogeneity contrast. Notably, our scheme operates entirely within an algebraic framework, eliminating the need for Dirichlet nodes and positive-definiteness on specific matrices arising in the model. This flexibility enables the simulation of a wider range of physical models and accommodates various boundary conditions. Rigorous theoretical analyses are provided under suitable assumptions, and extensive numerical experiments validate the effectiveness of the proposed approach.

math.NA↗

arXiv2026

Iterative Contact-resolving Hybrid Methods for Multiscale Contact Mechanics

Modeling contact mechanics with high contrast coefficients presents significant mathematical and computational challenges, especially in achieving strongly symmetric stress approximations for mixed formulations. Due to the inherent nonlinearity of contact problems, conventional methods that treat the entire domain as a monolithic system often lead to high global complexity. To address this, we develop an iterative contact-resolving hybrid method by localizing nonlinear contact constraints within a smaller subdomain, while the larger subdomain is governed by a linear system. Our system employs variational inequality theory, minimization principles, and penalty methods. More importantly, we propose four discretization types within the two-subdomain framework, ranging from applying standard/mixed FEM across the entire domain to combining standard/mixed multiscale methods in the larger subdomain with standard/mixed FEM in the smaller one. % The standard finite element method and standard constraint energy minimizing generalized multiscale finite element method are simple and easy to demonstrate. By employing a multiscale reduction technique, the method avoids excessive degrees of freedom inherent in conventional methods in the larger domain, while the mixed formulation enables direct stress computation, ensures local momentum conservation, and resists locking in nearly incompressible materials. Convergence analysis and the corresponding algorithms are provided for all cases. Extensive numerical experiments are presented to validate the effectiveness of the approaches.

math.NA↗

arXiv2025

A locking free multiscale method for linear elasticity in stress-displacement formulation with high contrast coefficients

Achieving strongly symmetric stress approximations for linear elasticity problems in high-contrast media poses a significant computational challenge. Conventional methods often struggle with prohibitively high computational costs due to excessive degrees of freedom, limiting their practical applicability. To overcome this challenge, we introduce an efficient multiscale model reduction method and a computationally inexpensive coarse-grid simulation technique for linear elasticity equations in highly heterogeneous, high-contrast media. We first utilize a stable stress-displacement mixed finite element method to discretize the linear elasticity problem and then present the construction of multiscale basis functions for the displacement and the stress. The mixed formulation offers several advantages such as direct stress computation without post-processing, local momentum conservation (ensuring physical consistency), and robustness against locking effects, even for nearly incompressible materials. Theoretical analysis confirms that our method is inf-sup stable and locking-free, with first-order convergence relative to the coarse mesh size. Notably, the convergence remains independent of contrast ratios as enlarging oversampling regions. Numerical experiments validate the method's effectiveness, demonstrating its superior performance even under extreme contrast conditions.

math.NA↗

arXiv2025

Efficient numerical method for the Schrödinger equation with high-contrast potentials

In this paper, we study the Schrödinger equation in the semiclassical regime and with multiscale potential function. We develop the so-called constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM), in the framework of Crank-Nicolson (CN) discretization in time. The localized multiscale basis functions are constructed by addressing the spectral problem and a constrained energy minimization problem related to the Hamiltonian norm. A first-order convergence in the energy norm and second-order convergence in the $L^2$ norm for our numerical scheme are shown, with a relation between oversampling number in the CEM-GMsFEM method, spatial mesh size and the semiclassical parameter provided. Furthermore, we demonstrate the convergence of the proposed Crank-Nicolson CEM-GMsFEM scheme. The convergence requires $H/\sqrtΛ=O(\varepsilon^{\frac{5}{4}})$, $Δt=O(\varepsilon^{\frac{5}{4}})$ if $\varepsilon\leq δ$; while if $δ<\varepsilon$, the convergence requires $H/\sqrtΛ=O(\varepsilon^{\frac{1}{4}}δ)$, $Δt=O(\frac{δ^2}{\varepsilon^{3/4}})$ (where $H$ represents the maximum diameter of coarse elements, $Λ$ is the minimal eigenvalue associated with the eigenvector not included in the auxiliary space, $Δt$ is the time step, $0 < \varepsilon\ll 1$ is the Planck constant and $δ$ describes the multiscale structure of the potential).Several numerical examples including 1D and 2D in space, with high-contrast potential are conducted to demonstrate the efficiency and accuracy of our proposed scheme.

math.NA↗

arXiv2023

Spectral analysis of a mixed method for linear elasticity

The purpose of this paper is to analyze a mixed method for linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise $(k+1)$, $k$ and $(k+1)$-th degree polynomial functions ($k\geq 1$), respectively. The numerical eigenfunction of stress is symmetric. By the discrete $H^1$-stability of numerical displacement, we prove an $O(h^{k+2})$ approximation to the $L^{2}$-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem, with proper regularity assumption. Thus via postprocessing, we obtain a better approximation to the eigenspace of exact displacement for the eigenproblem than conventional methods. We also prove that numerical approximation to the eigenfunction of stress is locking free with respect to Poisson ratio. We introduce a hybridization to reduce the mixed method to a condensed eigenproblem and prove an $O(h^2)$ initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete $H^1$-stability of numerical displacement, while only an $O(h)$ approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments.

math.NA↗

arXiv2023

Robust and flexible learning of a high-dimensional classification rule using auxiliary outcomes

Correlated outcomes are common in many practical problems. In some settings, one outcome is of particular interest, and others are auxiliary. To leverage information shared by all the outcomes, traditional multi-task learning (MTL) minimizes an averaged loss function over all the outcomes, which may lead to biased estimation for the target outcome, especially when the MTL model is mis-specified. In this work, based on a decomposition of estimation bias into two types, within-subspace and against-subspace, we develop a robust transfer learning approach to estimating a high-dimensional linear decision rule for the outcome of interest with the presence of auxiliary outcomes. The proposed method includes an MTL step using all outcomes to gain efficiency, and a subsequent calibration step using only the outcome of interest to correct both types of biases. We show that the final estimator can achieve a lower estimation error than the one using only the single outcome of interest. Simulations and real data analysis are conducted to justify the superiority of the proposed method.

stat.ME↗

arXiv2023

Efficient surrogate-assisted inference for patient-reported outcome measures with complex missing mechanism

Patient-reported outcome (PRO) measures are increasingly collected as a means of measuring healthcare quality and value. The capability to predict such measures enables patient-provider shared decision making and the delivery of patient-centered care. However, due to their voluntary nature, PRO measures often suffer from a high missing rate, and the missingness may depend on many patient factors. Under such a complex missing mechanism, statistical inference of the parameters in prediction models for PRO measures is challenging, especially when flexible imputation models such as machine learning or nonparametric methods are used. Specifically, the slow convergence rate of the flexible imputation model may lead to non-negligible bias, and the traditional missing propensity, capable of removing such a bias, is hard to estimate due to the complex missing mechanism. To efficiently infer the parameters of interest, we propose to use an informative surrogate that can lead to a flexible imputation model lying in a low-dimensional subspace. To remove the bias due to the flexible imputation model, we identify a class of weighting functions as alternatives to the traditional propensity score and estimate the low-dimensional one within the identified function class. Based on the estimated low-dimensional weighting function, we construct a one-step debiased estimator without using any information of the true missing propensity. We establish the asymptotic normality of the one-step debiased estimator. Simulation and an application to real-world data demonstrate the superiority of the proposed method.

stat.ME↗

arXiv2021

Real-Time Trajectory Planning for AGV in the Presence of Moving Obstacles: A First-Search-Then-Optimization Approach

This paper focuses on automatic guided vehicle (AGV) trajectory planning in the presence of moving obstacles with known but complicated trajectories. In order to achieve good solution precision, optimality and unification, the concerned task should be formulated as an optimal control problem, and then discretized into a nonlinear programming (NLP) problem, which is numerically optimized thereafter. Without a near-feasible or near-optimal initial guess, the NLP-solving process is usually slow. With the purpose of accelerating the NLP solution, a search-based rough planning stage is added to generate appropriate initial guesses. Concretely, a continuous state space is formulated, which consists of Cartesian product of 2D configuration space and a time dimension. The rough trajectory is generated by a graph-search based planner, namely the A* algorithm. Herein, the nodes in the graph are constructed by discretizing the aforementioned continuous spatio-temporal space. Through this first-search-then-optimization framework, optimal solutions to unified trajectory planning problems can be obtained fast. Simulations have been conducted to verify the real-time performance of our proposal.

cs.RO↗

arXiv2013

Growth Patterns of US Children from 1963 to 2012

Anthropometric measurements such as weight, stature (height), and body mass index (BMI) provide reliable indicators of children's growth. The 2000 CDC growth charts are the national standards in the United States for these important measures. But these growth charts were generated using data from 1963-1994. To understand the growth patterns of US children since 1994, we generate weight-for-age, stature-for-age and BMI-for-age percentile curves for both boys and girls aged 2-20 through the methods used to generate the 2000 CDC growth charts. Our datasets are from the National Health and Nutrition Examination Survey (NHANES) for years 1999-2010 and and from NorthShore University HealthSystem's Enterprise Data Warehouse (NS-EDW) for years 2006-2012. The weight and BMI percentile curves generated from NS-EDW and NHANES data differ substantially from the CDC percentile curves, while those for stature do not differ substantially. We conclude that the population weight and BMI values of US children in recent years have increased significantly since 2000 and the 2000 CDC growth charts may no longer be applicable to the current population of US children. Our charts poignantly reveals the increasing obesity of American children.

stat.AP↗